+1 on what MREJ says - it's similar to the Direct Route from SOP to East Hills on Ride The City:
http://www.ridethecity.com/#2145170 I prefer to use East St at Lidcombe (rather than Vaughan St), then head west at Georges Ave
towards Campbell St Berala as there are less speed bumps and less contention with traffic.
Safest Route on RTC is via the Liverpool Rail Trail :
http://www.ridethecity.com/#2145169 , but I'm not sure about the
section on Henry Lawson Drive south of Newbridge Rd - it's heavily trafficked by trucks etc.., in peak hour that are
(IIRC) fairly free flowing. And there's also some hairy bits around the railway stations in peak hours.
The safer route on RTC
http://www.ridethecity.com/#2145177 shows another option that goes by Dunc Gray velodrome,
and would be a bit more relaxed than the William St/Edgar St/Queen St option shown in the direct route. The section
of Henry Lawson Drive north of Newbridge Rd is usually bumper to bumper in peak hour and has a good enough shoulder
to ride around from the overbridge (near Marsden) to Rabual Rd where the decent cycleway starts on the western
side of HLD.